Relations and Functions

Relations

Relations and functions map a domain (x values) to a range (y values). A function is a special type of relation.

Relations describe the manner in which an independent variable maps to a dependent variable.

A line is a relation between any x value and a single y value in the Cartesian coordinate system. The data could also be represented by a table with two columns. The pairs of values are called 'Ordered Pairs': the Domain (input) is the largest set of values which map to the Range (output or image) set of values.

A relation is a function $f$ if:

- $f$ acts on all elements of the domain

- $f$ pairs each element of the domain with one and only one element of the range

In Cartesian graphs of relations and functions, traditionally $y = f(x)$. The relation can also be expressed as $f: x ↦ f(x)$, where $f$ is a function that maps $x$ into $f(x)$.

The independent variable is $x$ (also known as the argument of the function), and the dependent variable is y.

Functions

A function is a correspondence (mapping) between two sets X and Y in which each element of set X maps to (corresponds with) exactly one element of set Y.

Vertical line test

A relation is a function if a vertical line drawn anywhere on a graph of the relation does not cross the curve more than once.

e.g. a vertical line drawn through $f(x) = x^2$ crosses no more than once (function), but $f(x) = ±√x$ has two intersections (not a function).

A vertical line parallel to the y-axis has more than one (in fact infinite!) y-values for one x-value. A vertical line is a relation but not a function.

A horizontal line parallel to the x-axis has the same y-value for any value of x. It is a relation AND a function.

Graph of the quadratic $y = x^2$

The relation $y = x^2$ has one y-value for two x-values, except at the minimum or maximum, where there is one for one. This is a function, so may be expressed as $f(x) = x^2$.

Graph of the relation $y = ±√x$

The relation $y^2 = x$, however, has two y-values for each x, so is not a function.

Asymptotes and Discontinuities

Now consider the equation: $g(x) = 1/{(x^2-1)}$

What are its domain and range?

If the denominator of a fraction is zero, there is no solution. As the denominator approaches zero, the value of the function gets very large (tends to infinity).

This function therefore has three asymptotes: x = -1, x = +1, y = 0.

Domain: ]-∞, -1[ ∪ ]-1, +1[ ∪ ]+1, +∞[

The graph is discontinuous for values of g(x) between -1 and 0.

Range: ]-∞, -1] ∪ ]0, +∞[

Even and odd functions

A function is even, if for all $x$ in the domain, $-x$ is in the domain, and $f(x) = f(-x)$ for all values of $x$.

Even functions are symmetrical about the y-axis. cos$(x)$ is an example of an even function.

A function is odd, if for all $x$ in the domain, $-x$ is in the domain, and $f(-x) = -f(x)$ for all values of $x$.

Odd functions are NOT symmetrical about the y-axis. sin$(x)$ is an example of an odd function.

one-to-one and one-to-many

Remember that a relation which has more than one value of y for each value of x is not a function.

However, a y value may have more than one value of x. This is a one-to-many relationship. An example is $y = x^2$.

If there is only one value of y for each value of x, such as in $y = x$, then the function is one-to-one.

A horizontal line test will reveal whether there is more than one x value for one y value.